It seems like you're wondering if it's worth to buy the 'best gear' to start out with or if there's a point to doing a progression.

Shoes: If you're starting out, get the cheapest shoes that fit. Your crappy footwork (there is never an exception) will wear the soles out in no time. Use them until the toe is about to blow out and then consider upgrading (by that time, you will have improved enough that you won't wear them out as quickly).

Belay Device: You should definitely get one of the mainstream tube devices to start. Either a BD ATC XP (or ATC Guide) or a Petzl Reverso. Become very familiar with it.

Later on you might want to check out the Mammut Smart Alpine and the Petzl Grigri depending on what kind of climbing you gravitate towards. Don't start using these from the get-go since you'll build bad habits.

Regardless, belay devices are pretty cheap so buying a bunch over the long haul isn't a big deal.

Carabiners: Buy a high quality locking HMS 'biner for your belay 'biner. Don't start off using one of those special belay 'biners with the tail that keeps the rope in place (it'll breed bad habits). With regular climbing, it will wear out.

Chalk Bag: Get something that looks cool and works for you. Zero performance improvements to be gained on this front (maybe weight).

Harness: Get a good harness but don't pay way too much. The basic BD Momentum harness will do everything you need and when you do upgrade you'll have it around in case a friend needs to climb. They'll last a REALLY long time (just make sure to inspect the tie in points, etc. for wear).

Quickdraws: You won't need them if you're toproping in a gym, but when you start leading you will. I'm going to differ from popular opinion here, but get the lightest draws you can afford. They may not be the most ergonomic, but you'll grow used to them and the 'biners will then be usable as alpine draws when you go trad climbing. You'll want to get 15-18 draws (BD draws come in sets of 6, many others come in sets of 5).

DO NOT BUY QUICKDRAWS UNTIL YOU START LEADING.

Once you get to the point where you're considering leading on gear, you'll find plenty of threads on how to choose a first rack. Until then, this should be plenty.

There are areas that require 15-18 draws for a single pitch of sport climbing, but unless you live close to one of these places 10-12 will be a good start(in a pinch you and your climbing partner can pool your gear to get through a route). You could very likely have 20 draws someday, particularly if you want a separate set of alpine draws for trad, but no need to buy everything at once.

If possible borrow/try out your partners' gear in the beginning. It sucks to drop a lot of money on a set of draws, rack, etc only to realize that you don't like that brand/model.